estimation is a powerful tool for analyzing time series data. It breaks down complex signals into their frequency components, revealing hidden patterns and that might not be apparent in the raw data.
Choosing between parametric and involves trade-offs in accuracy and flexibility. help balance the , allowing analysts to tailor their approach to the specific characteristics of their data.
Spectral Density Estimation
Concept of spectral density
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Spectral density, also known as (PSD), describes the distribution of power or across different frequencies in a time series
Provides information about the relative importance of different frequency components (low frequency, high frequency) in the series
Represents the decomposition of the time series into a sum of sinusoidal components with different frequencies and amplitudes
of the of a stationary time series
Analyzing spectral density allows for identifying or periodicities (seasonal patterns, business cycles), detecting hidden patterns or cycles, and understanding the relative contributions of different frequency components to the overall variability of the series
Parametric vs non-parametric estimation methods
assume a specific model for the time series (autoregressive (AR), moving average (MA)) and estimate the parameters of the assumed model to calculate the spectral density
Examples: , Burg method
Advantages: provide smooth spectral density estimates, require fewer data points
Disadvantages: rely on correct model specification, may not capture complex or non-linear relationships
Non-parametric methods do not assume a specific model and estimate the spectral density directly from the data using techniques like periodogram or smoothed periodogram
Examples: , Welch method
Advantages: do not require assumptions about the underlying model, can capture complex or non-linear relationships
Disadvantages: may produce noisy or less smooth estimates, require more data points
Smoothing techniques for spectral density
Smoothing techniques reduce variance and improve stability of spectral density estimates
multiplies the time series by a window function (Hamming, Hann, Bartlett) before computing the periodogram
Reduces and improves resolution of the estimate
divides the time series into overlapping or non-overlapping segments, computes the periodogram for each segment, and averages the periodograms
Reduces variance at the cost of reduced frequency resolution